Wednesday, June 23, 2010

Science of Doom discussed recent theorizing on the causes of high temperatures on Venus, as did Chris Colose. On both blogs, Leonard Weinstein proposed a thought experiment in which a shell was placed over Venus - he believed that the high surface temperatures would remain. SoD took up this idea here, and there was further discussion. This post sets out my ideas on the problem, with some background.

Heat transfer mechanisms

Conduction

Radiation

Transmission of thermal range IR in an atmosphere like Venus is not like transmission in a vacuum. At most wavelengths the absorption length L is fairly small relative to the depth of the atmosphere. However, heat balance requires that absorption is balanced by emission. This gas-to-gas transmission is very dependent on the temperature gradient.

The Rosseland model of IR transmission works for high opacity gas. If you simplify by neglecting scattering, then
F/F0 = (16/3)*(L/*T0) ∇ T
where F0 is BB (black body) emission at ambient T0, F is flux.
This is a Fourier Law with a conductivity that can be quite high. For example, if L is 1 km, T0=400K, the flux is about 0.13 times BB flux.

You can visualise how this transmission works. Imagine yourself with IR vision in such an atmosphere with a lapse rate. The gas below is hotter than the gas above. If you hold out your hand, it's warmer below than above. There is a nett heat flux upward.

How much? Well, it's proportional to the lapse rate, which determines the temperature difference that you see. But it depends on the absorption length too. If that is higher, you can see further, and see hotter (and colder) gas, because of the linear lapse rate. This effect is proportional to L, hence the Rosseland expression.

As L increases further, things change. Looking down, the gas gets denser, so you can't see as far. The hotness increases less than linearly. Looking up, you see further. The coolness increases more than linearly. They cancel somewhat, so changes to the flux are second order in L. But eventually there is a deviation from Fourier's Law. And if you see far enough, the ground will have an effect too.

Eventually, when L is large relative to the depth of the troposphere, a substantial fraction of IR is transmitted without any absorption. This is an atmospheric window, and the relation between transmission and lapse rate diminishes.

But up to that point, Fourier's Law is a good way of thinking about IR transmission.

Convection

This also has a wrinkle. On a room scale, turbulent convection is also often thought of as obeying a Fourier Law. Gravity-forced changes of pressure are negligible. The adiabatic lapse rate is about 0.01 C/m.

However, on the atmospheric scale, compression with gravity is important in modifying the temperature. Compression warms. Fourier's Law applies not to the gradient of temperature, but of potential temperature:
θ=T(P/P0)^(-ν)
where ν=R/cp and P0 is a reference pressure.

The important thing here is that unlike with conductive and radiative transport, convective transport is zero at the dry adiabat lapse rate, rather than zero gradient, and is proportional to the difference between the actual lapse rate and the adiabat. Below the adiabat, the gas is convectively stable, and energy is used making it go up and down. Above the adiabat, it is unstable, and the temperature gradient adds energy to the motion, and accelerates the transport.

Atmospheric fluxes

Now that we've outlined the modes, and agreed to ignore conduction, we have a downflux of sunlight on Venus, after allowing for albedo, of about 160 W/m2. Not much of that actually reaches the surface - it is absorbed at various depths. But once absorbed, the heat has to get out again, and the modes available for it to reach the tropopause to be radiated away are IR transfer and convection.

Suppose the convective component were small. The lapse rate would then be determined by a Fourier Law, as given by the Rosseland model. This might be more or less than the adiabat.

If more, then the gas would be convectively unstable. Motions would be induced which would increase convective transport. Since the overall flux is determined by the sunlight, that means the proportion carried by IR would reduce, lowering the lapse rate. This argument, made by DeWitt in the first SoD thread, gives a mechanism whereby the adiabat lapse rate cannot be much exceeded.

But there is an analogous argument from below. If the IR flux caused a temperature gradient less than the adiabat, then the gas would be convectively stable, and it could remain at that. However, there are likely to be other effects inducing motion. Polar regions (and the long nights on Venus) emit more heat than they receive - sunny regions emit less. This heat has to be transported by the atmosphere, and the temperature difference drives a heat engine. The resulting circulation conveys the heat.

But the motion then affects vertical convection. I described earlier how motion pumps heat downward. This then augments the heat that must come up conveyed by IR. This in turn increases the lapse rate towards to adiabat. In doing so, it extracts KE from the air to drive the pump.

Leonard Weinstein's problem

At SoD, Leonard Weinstein proposed a thought experiment where a shell was placed around Venus at about the altitude where IR is currently emitted to space. This is high in the troposphere, where the temperature is about 230K. The shell is opaque to all radiation, but a good conductor. He contended that the temperature profile in the atmosphere would remain much the same. My initial thought was that it would become isothermal, as did others.

I now think a reasonably quantitative analysis is possible. About 160W/m2 sunlight penetrates the atmosphere, and is balanced by an IR and a convective flux. I don't know how much is radiative, but the total is known.

The shell blocks that 160W/m2. If the atmosphere remains the same, then the same IR transport upwards must continue, because it is determined by the unchanged temperature gradient. The convection would also continue, since it is determined by the gradient and the motion. But there's no longer heat supplied, so something has to change. Net upflux must go to zero.

The lapse rate must drop well below the adiabat, to ensure that convective flow is downward, by the heat pumping mechanism. If it does, there will be a balance when the downflux matches the IR upflux, giving the required zero. But how low would the lapse rate go?

Heat pump arithmetic

In this scenario, the heat pump must pump downward about 160 W/m2 than it did before the shell. It has to pump it to depths comparable to where sunlight formerly penetrated. Figures get rough here, but let's say the average depth is where the temperature has risen from 230K to 460K.

Thermodynamics tells us the energy needed to do that. It's 160*(δ T)/T, or about 80 W/m2. That comes from the KE in the gas, which must in turn have come from a heat engine somewhere.

But the equator to pole differential, which is a promising source of energy for a heat engine, is much too small. The actual flux of heat redistributed horizontally can only be a small fraction of the 160 W/m2 arriving. And the temperature differential, even with reduced circulation, is never going to match the factor of 2 difference between surface and depth ( without a shell, it's very small).

In fact, there just isn't that sort of energy available anywhere. It's half of total average incoming solar.

So what would happen?

If the convective heat pump can't replace the blocked sunlight, the lapse rate must just reduce until the IR Fourier's Law flux can be matched by the very limited energy available from a realistic heat engine. This needs a rather elaborate but doable calculation which would consider differential insolation over a sphere, and a plausible temperature differential around the shell. The resulting regional discrepancy between sunlight in and IR out would determine the horizontal fluxes, and the temperatures would determine the energy available as KE to the atmosphere. Then some unknown fraction of that could pump heat down, making possible an IR upflux and a positive lapse rate.

So a SWAG? Dunno, but surface temperatures warmer than the shell, but Earth-like rather than Venus-like.

Saturday, June 19, 2010

Steven Mosher was looking through the code for TempLS v1.4 when he saw something puzzling and asked me about it. And sure enough, it was an error. One of those things a fresh pair of eyes can notice.

It was in the section that takes the station list and assigns them to a cell number, depending on what 5x5 lat/lon block they fall into. It doesn't matter what the number is, as long as the stations in the block have the same number and others have different numbers. The stations with that number are counted, and the sum is used in the least squares weighting.

The first line divides the lat/lon by 5 and rounds to an integer. So each station has an integer pair identifying its cell.

They range from (-18,-36) to (17,35) - there are 2592 cells.

The next line tries to number these from 1 to 2592. Numbering a rectangle array is normally done either by row or by column. The 1333 is an offset to bring them to a positive range. I'm numbering in rows, so 36 should be the number of columns. But alas, there are 36 rows and 72 columns. I made the mistake because lat/lon, on the map, is actually in (y,x) order rather than the conventional (x,y).

I'll post the revised code at Ver 1.41. Ver 2.0 should be out soon.

What is the effect?

Basically, two cells get assigned to every number. It turns out that they are opposite in longitude. This affects the weighting, which is meant to correct for station concentration. When an area is well covered by stations, the least squares process downweights each individual station, so the area does not get undue attention. But by combining with a cell on the other side of the world, this downweight could be out by a factor of two.

I should hastily say that it doesn't mean that temperatures from the other side of the world are modifying the readings. They don't - they only modify the weight given to the readings.

Fortunately, it won't matter for any regional studies. The reason is that there are then no stations included from the other side of the world. The weights are unaffected.

A digression on weighting

Another way of seeing the use of weighting by inverse density is that it does what you would do if you were trying to use the stations in a numerical integral expression. As such. it has a fault when grid cells can be empty. The natural thing to do with an empty cell is just to leave it out, as GISS does, and I suspect the other indices. So did I. But leaving it out, when compiling a global average, is effectively the same as including it with a value artificially assigned to it, which is equal to the global average. And when you look at spatial patterns of anomalies etc, it's clear this often isn't optimal. You might have regions like the Arctic where the cells that do have stations have high (or low) values, but many are missing. Implicitly assigning these to have average value biases the result down (or up).

GIStemp get criticised for extrapolating over wide regions in the Arctic. But it's the best thing to do (in a bad situation). leaving the regions out seems more conservative, but it isn't.

Anyway I've been looking at ways of using irregular triangular subdivisions instead of a regular grid. The idea is that when stations get sparse, you just make the subdivisions bigger. No part of the Earth lacks a representative cover. As I mentioned, sparseness is a problem - here the triangles get big, so the coverage degrades. But it's better than none.

Back to the error

I recalculated some of the main global plots of recent posts. Here old and new are contrasted. As you'll see, the discrepancies are noticeable but not major. Mainly the error made peaks and valleys more extreme. I don't believe the revised data would show out on Zeke's combined blogger plots. At the bottom of each plot the modified trend is shown. The changes are small.

There are sometimes some biggish changes in late 1930's. I think this reflects the fact that these years were hot in ConUS, which were then overweighted in the older version.

Sunday, June 6, 2010

I have been collecting global surface data series for comparison with models. In the process I've made a big table, which is handy for reading into R. It contains:
1. The surface land and ocean indices - Hadcrut3, GIStemp and NOAA (current to April 2010)
2. The two satellite lower trop indices - UAH and MSU RSS (likewise)
3. A collection of model results for surface air temp (SAT). There are 24 models, using SRES A1B (scenario), and a total of 57 runs.
The table goes from 1850 to 2300, although of course most columns have blanks before and after the real data.